Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Agarwal, A. and Claisse, J. (2020) Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method. Stochastic Processes and their Applications, 130(8), pp. 5006-5036. (doi: 10.1016/j.spa.2020.02.009)

Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity in bounded domains and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. We consider several examples and estimate their solution by using the Monte Carlo method.

• Item Type: Articles
• Keywords: Branching diffusion processes, partial differential equation, semi-linear, elliptic, Monte Carlo method, automatic differentiation formula.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Agarwal, Dr Ankush
• Authors: Agarwal, A., and Claisse, J.
• Subjects: Q Science > QA Mathematics
• College/School: College of Social Sciences > Adam Smith Business School > Economics
• Journal Name: Stochastic Processes and their Applications
• Journal Abbr.: SPA
• Publisher: Elsevier
• ISSN: 0304-4149
• ISSN (Online): 1879-209X
• Published Online: 28 February 2020
• Copyright Holders: Copyright © 2020 Elsevier B.V.
• First Published: First published in Stochastic Processes and their Applications 130(8): 5006-5036
• Publisher Policy: Reproduced in accordance with the publisher copyright policy